Iterative wavefront tailoring to simplify freeform optical design for prescribed irradiance


Compared with traditional spherical and aspherical optics, freeform optics has more degrees of freedom which can better reshape the beam in a prescribed manner. However, the design formulation of a freeform reflective or refractive surface for producing prescribed irradiance patterns from a point source has been a great challenge. By reducing the design formulation to fully nonlinear partial differential equations that can be solved through Newton’s methods, a direct determination of the freeform optical surfaces has been made possible. This can be further attributed to substituting the ray -trace equations into energy conservation equations by taking into consideration the surface integrability condition. Even though this approach presents the design problem in one equation, the formula derivation process is highly complex and complicated.

The aforementioned problem has led to the development of several solution methods for the prescribed irradiance problem from a point source. The ray-mapping method is widely used. It’s based on approximating the ray map between the source and target irradiance distributions which are used in constructing the freeform optical surfaces to reduce the design complexity. Unfortunately, obtaining an exact ray map is very difficult due to its connection with the optical element geometry.

Recently, Beijing Institute of Technology researchers Assistant professor Zexin Feng, Professor Dewen Cheng, and Professor Yongtian Wang from the School of Optics and Photonics developed two new iterative methods to simplify the freeform optical design based on the outgoing beam propagation properties. The main objective was to simplify the design process as compared to the conventional direct determination approach.

The first is an iterative wavefront tailoring method. Fundamentally, a general partial differential equation was set by connecting the properties of the outgoing wavefront to energy conservation. The wavefront equation solution generated a ray map that was used to construct an approximate freeform optical element. Using the data generated by the optical element, the wavefront equation was updated to obtain a more accurate ray map to enhance its performance. Additionally, the process was further iterated to reduce surface errors and enhance irradiance accuracies. Even though the iterative wavefront tailoring method is applicable for reflective designs, this paper was mainly focused on refractive designs. The work was published in the research journal, Optics Letters.

The second is an intermediate irradiance transport method.  The key point lies in that the source irradiance before the required freeform lens was transported into an intermediate irradiance estimate immediately behind the lens. This transport is simply realized by applying a collinear relationship among the coordinates on the intermediate plane, exit freeform surface and target plane to energy conservation. In this way, one can solve a phase retrieval problem from the intermediate and target irradiance distributions to acquire a ray map. Similar with the iterative wavefront tailoring method, an iterative ray map computation and surface construction is also used to improve the performance, but the resulting phase equation is simpler than the parametric wavefront equation. The work is currently published in the research journal, Optics Letters.

 The general equation of a parametrized outgoing wavefront and the phase equation were derived without taking into consideration the optical element structure. The new methods comprise a simplified formula derivation process and the abilities to generate a variety of freeform optical structures with high accuracy as compared to the direct determination method. On the other hand, the implemented iterative procedures may solve the unique numerical instabilities associated with the Newton method through a new numerical algorithm approach. Furthermore, a multi-scale strategy can be employed to implement the two iterative procedures, which can speed up the computations.

Many different designs were developed to prove the effectiveness and flexibility of the proposed method. This included the design of a spherical freeform lens and that of a plano-freeform structure. Besides, the two methods are suitable for tackling designs with freeform entrance surface that have been challenging in the past.  In general, the two methods significantly simplified the freeform lens design for a prescribed irradiance problem by utilizing the outgoing beam propagation properties. Whereas they are only applicable to smooth optical surfaces, Beijing Institute of Technology scientists who led the research are optimistic that the restriction will facilitate its fabrication. Altogether, the study provides essential information that will advance the development of more robust numerical solution methods for controlling both the irradiance and wavefront of laser beams.  The two methods are valuable contributions to the state of the art, and the illumination community may benefit a lot from them.

complicated freeform optical beam shaping-Advances in Engineering

complicated freeform optical beam shaping-Advances in Engineering

A plano-freeform lens prototype generates letters “FREEFORM OPTICS” from a normal LED source. Grid spacing: 0.5 mm for the left photograph. The lens was designed by the intermediate irradiance transport method.

About the author

Zexin Feng currently focuses on freeform optics design for illumination, beam shaping and imaging applications. He received his Ph.D. in Electronic Science and Technology from Tsinghua University in June, 2012. He stayed as a post-doctor in the field of Optical Engineering at Tsinghua University until April, 2014. Afterwards, he went to the USA and performed two years of postdoctoral work at College of Optical Sciences, University of Arizona. He is now an assistant professor at School of Optics and Photonics, Beijing Institute of Technology.

About the author

Dewen Cheng is a professor of optics in Beijing Institute of Technology (BIT). He got his PhD from School of Optics and Photonics, Beijing Institute of Technology in June 2011. He was a joint PhD student at College of Optical Sciences, University of Arizona from August 2007 to August 2010. His research interests lie in optical design, optical simulation, freeform optics and near eye display. He has published more than 70 peer-reviewed papers in international and national well-known optical journals and conferences. Dr. Cheng was selected as one of the New Century Talents supported by the Chinese Ministry of Education in 2012. He was also selected as one of the Top Young Talents in the National Ten-thousand Talents Program in 2014. He has been supported by the National Science Fund of China for Excellent Young Scholars since 2018.

About the author

Yongtian Wang is a professor of optics and applied computer science at Beijing Institute of Technology (BIT), and a Chang Jiang Scholar appointed by the Ministry of Education of China. He is the founding director of Beijing Engineering Research Center of Mixed reality and Advanced Display. His research interests include optical design and CAD, optical instrumentation, 3D display, image processing, virtual reality and augmented reality, and human-computer interaction. He is a Fellow of SPIE, OSA and IET, and a member of the board of directors of the Chinese Society for Optical Engineering (CSOE), China Computer Federation (CCF) and China Society of Image and Graphics (CSIG).


Feng, Z., Cheng, D., & Wang, Y. (2019). Iterative wavefront tailoring to simplify freeform optical design for prescribed irradiance. Optics Letters, 44(9), 2274.

Go To Optics Letters

Zexin Feng, Dewen Cheng, and Yongtian Wang (2019). Transferring freeform lens design into phase retrieval through intermediate irradiance transport. Optics Letters 44 (22), pp. 5501-5504.

Go To Optics Letters

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